Simplify the expression \$((2x^3y^2)(-3xy^4)^2) / (6x^4y^3)^2\$.

Given expression is

\$((2x^3y^2)(-3xy^4)^2) / (6x^4y^3)^2\$

First, let’s simplify each part individually

\$(2x^3y^2) = 2x^3y^2\$

\$(- 3xy^4)^2 = (-3)^2 times (x)^2 times (y^4)^2 = 9x^2y^8\$

\$(6x^4y^3)^2 = (6)^2 times (x^4)^2 times (y^3)^2 = 36x^8y^6\$

Now plug these values back into the expression

\$((2x^3y^2)(9x^2y^8) )/ (36x^8y^6)\$

Combine like terms by multiplying coefficients and adding exponents of variables

\$(2 times 9 times x^3 times x^2 times y^2 times y^8) / (36 times x^8 times y^6)\$

\$(18x^5y^10) / (36x^8y^6)\$

\$(x^5y^10) / (2x^8y^6)\$

After simplification

\$(1/2) times (x^5 / x^8) times (y^10 / y^6)\$

\$(1/2) times x^(5 - 8) times y^(10 - 6)\$

\$(1/2) times x^(- 3) times y^4\$

\$(1/2) times y^4 / x^3\$

Therefore, the simplified expression is \$(1/2) times y^4 / x^3\$.

Option A inaccurately presents \$−(y^4)/(2x^3)\$, due to the inclusion of a negative sign, rendering it incorrect

\$ - y^4 / (2x^3)\$

It simplifies to \$y^3/(2x^2)\$, differing from our derived simplified form, thus incorrect

\$ y^3 / (2x^2)\$

Correct: It precisely performs the calculations with accuracy

\$ (1/2) times y^4 / (x^3)\$

\$−(y^3)/(2x^2)\$, is invalid as it diverges from the simplified expression due to calculation errors

\$ (- y^3) / (2x^2)\$

Option

C

Can you summarize what you’ve understood in the above steps?